Download
1 / 66
E N D
Syllabus Lecture 01 Describing Inverse ProblemsLecture 02 Probability and Measurement Error, Part 1Lecture 03 Probability and Measurement Error, Part 2 Lecture 04 The L2 Norm and Simple Least SquaresLecture 05 A Priori Information and Weighted Least SquaredLecture 06 Resolution and Generalized Inverses Lecture 07 Backus-Gilbert Inverse and the Trade Off of Resolution and VarianceLecture 08 The Principle of Maximum LikelihoodLecture 09 Inexact TheoriesLecture 10 Nonuniqueness and Localized AveragesLecture 11 Vector Spaces and Singular Value Decomposition Lecture 12 Equality and Inequality ConstraintsLecture 13 L1 , L∞ Norm Problems and Linear ProgrammingLecture 14 Nonlinear Problems: Grid and Monte Carlo Searches Lecture 15 Nonlinear Problems: Newton’s Method Lecture 16 Nonlinear Problems: Simulated Annealing and Bootstrap Confidence Intervals Lecture 17 Factor AnalysisLecture 18 Varimax Factors, Empirical Orthogonal FunctionsLecture 19 Backus-Gilbert Theory for Continuous Problems; Radon’s ProblemLecture 20 Linear Operators and Their AdjointsLecture 21 Fréchet DerivativesLecture 22 Exemplary Inverse Problems, incl. Filter DesignLecture 23 Exemplary Inverse Problems, incl. Earthquake LocationLecture 24 Exemplary Inverse Problems, incl. Vibrational Problems
Purpose of the Lecture • Discuss how an inexact theory can be represented • Solve the inexact, linear Gaussian inverse problem Use maximization of relative entropy as a guiding principle for solving inverse problems Introduce F-test as way to determine whether one solution is “better” than another
How do we generalize the case ofan exact theoryto one that is inexact?
exact theory case model,m dobs dpre theory d=g(m) datum,d map mest
to make theory inexact ... must make thetheory probabilisticor fuzzy model,m dobs dpre d=g(m) datum,d map mest
a prior p.d.f. theory combination model,m model,m model,m dobs dobs dobs dpre datum,d datum,d datum,d map map map mest
how do youcombinetwo probability density functions ? so that the information in them is combined ...
desirable properties order shouldn’t matter combining something with the null distribution should leave it unchanged combination should be invariant under change of variables
a priori , pA theory, pg total, pT model,m model,m model,m dobs dobs dobs dpre datum,d datum,d datum,d map map map mest (D) (E) (F) model,m model,m model,m dobs dobs dobs dpre datum,d datum,d datum,d map map map mest
“solution to inverse problem”maximum likelihood point of (withpN∝constant) simultaneously gives mest and dpre
probability that the estimated model parameters are near m and the predicted data are near d T probability that the estimated model parameters are near m irrespective of the value of the predicted data
conceptual problem and T do not necessarily have maximum likelihood points at the same value of m
model,m dobs dpre datum,d map mest p(m) model,m mest’
illustrates the problem in defining a definitivesolution to an inverse problem
illustrates the problem in defining a definitivesolution to an inverse problem fortunately if all distributions are Gaussian the two points are the same
Part 2Solution of the inexact linear Gaussian inverse problem
Gaussian a priori information a priori values of model parameters their uncertainty
Gaussian observations observed data measurement error
Gaussian theory linear theory uncertainty in theory
mathematical statement of problem find (m,d) that maximizes pT(m,d) = pA(m) pA(d) pg(m,d) and, along the way, work out the form of pT(m,d)
notational simplification group m and d into single vector x = [dT,mT]T group [covm]A and [covd]A into single matrix write d-Gm=0 asFx=0 with F=[I, –G]
after much algebra, we findpT(x) is a Gaussian distributionwith mean and variance
after much algebra, we findpT(x) is a Gaussian distributionwith mean and variance solution to inverse problem
after pulling mest out of x* reminiscent of GT(GGT)-1 minimum length solution
after pulling mest out of x* error in theory adds to error in data
after pulling mest out of x* solution depends on the values of the prior information only to the extent that the model resolution matrix is different from an identity matrix
and after algebraic manipulation which also equals reminiscent of (GTG)-1 GT least squares solution
interesting aside weighted least squares solution is equal to the weighted minimum length solution
what did we learn? for linear Gaussian inverse problem inexactness of theory just adds to inexactness of data
Part 3Use maximization of relative entropy as a guiding principle for solving inverse problems
assessing the information contentin pA(m) Do we know a little about m or a lot about m ?
Information Gain, S • -S called Relative Entropy
(A) pA(m) pN(m) m (B) S(σA) σA
Principle ofMaximum Relative Entropyor if you prefer Principle ofMinimum Information Gain
find solutionp.d.f.pT(m) that has the largest relative entropy as compared to a priori p.d.f. pA(m) • or if you prefer • find solutionp.d.f.pT(m) that has smallest possible new information as compared to a priori p.d.f. pA(m)
properly normalized • p.d.f. • data is satisfied in the mean • or • expected value of error is zero
After minimization using Lagrange Multipliers process pT(m) is Gaussian with maximum likelihood point mest satisfying
After minimization using Lagrane Multipliers process pT(m) is Gaussian with maximum likelihood point mest satisfying just the weighted minimum length solution
What did we learn? Only that the Principle of Maximum Entropy is yet another way of deriving the inverse problem solutions we are already familiar with
Part 4F-testas way to determine whether one solution is “better” than another
